arXiv subjects

Wolter Groenevelt

Publications and source records attributed to Wolter Groenevelt.

31 records · Page 2Linked to original sources

arXiv2012

Spectral decomposition and matrix-valued orthogonal polynomials

The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as $2\times 2$-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.

math.CA↗

arXiv2011

Quantum Analogs of Tensor Product Representations of su(1,1)

We study representations of $U_q(su(1,1))$ that can be considered as quantum analogs of tensor products of irreducible *-representations of the Lie algebra $su(1,1)$. We determine the decomposition of these representations into irreducible *-representations of $U_q(su(1,1))$ by diagonalizing the action of the Casimir operator on suitable subspaces of the representation spaces. This leads to an interpretation of the big $q$-Jacobi polynomials and big $q$-Jacobi functions as quantum analogs of Clebsch-Gordan coefficients.

math.QA↗

arXiv2010

The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C)

The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U_q(su(1,1)). It turns out that U_q(su(1,1)) does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analogue of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to U_q(su(1,1))-representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. The paper is split into two parts; the first part gives almost all of the statements and the results, and the statements in the first part are independent of the second part. The second part contains the proofs of all the statements.

math.QA↗

arXiv2009

The indeterminate moment problem for the $q$-Meixner polynomials

For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous $q^{-1}$-Hahn and $q$-Laguerre polynomials.

math.CA↗

arXiv2007

Multivariable Wilson polynomials and degenerate Hecke algebras

We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials.

math.RT↗

arXiv2007

The vector-valued big q-Jacobi transform

Big $q$-Jacobi functions are eigenfunctions of a second order $q$-difference operator $L$. We study $L$ as an unbounded self-adjoint operator on an $L^2$-space of functions on $\mathbb R$ with a discrete measure. We describe explicitly the spectral decomposition of $L$ using an integral transform $\mathcal F$ with two different big $q$-Jacobi functions as a kernel, and we construct the inverse of $\mathcal F$.

math.CA↗

arXiv2005

Fourier transforms related to a root system of rank 1

We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a $q=1$ analogue of Sahi's double affine Hecke algebra related to the affine root system of type $(C^\vee_1, C_1)$. We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra $\mathcal H$, and the corresponding Fourier transforms. These eigenfunctions are non-symmetric versions of the Wilson polynomials and the Wilson functions.

math.RT↗

arXiv2005

Wilson function transforms related to Racah coefficients

The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for $U_q(\su(1,1))$, which turn out to be Askey-Wilson functions and Askey-Wilson polynomials.

math.CA↗

arXiv2003

The Wilson function transform

Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.

math.CA↗

arXiv2003

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra $\mathfrak{su}(1,1)$. The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of $\mathfrak{su}(1,1)$ are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by a discontinuous integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.

math.CA↗

arXiv2003

Continuous Hahn functions as Clebsch-Gordan coefficients

An explicit bilinear generating function for Meixner-Pollaczek polynomials is proved. This formula involves continuous dual Hahn polynomials, Meixner-Pollaczek functions, and non-polynomial $_3F_2$-hypergeometric functions that we consider as continuous Hahn functions. An integral transform pair with continuous Hahn functions as kernels is also proved. These results have an interpretation for the tensor product decomposition of a positive and a negative discrete series representation of $\su(1,1)$ with respect to hyperbolic bases, where the Clebsch-Gordan coefficients are continuous Hahn functions.

math.CA↗

arXiv2002

Bilinear summation formulas from quantum algebra representations

The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q^{-1}, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of 2-phi-1 -series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.

math.QA↗

arXiv2001

Meixner functions and polynomials related to Lie algebra representations

The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra su(1,1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and the discrete terms are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch-Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.

math.CA↗